Permittivity dispersion

Fig. 2.37 Permittivity dispersion and dielectric loss for a glass 18 Na2O-10CaO-72SiO2 (after Taylor, H.E., J. Soc. Glass Tech., 43, 124T, 1959. Also see Rawson, H. Properties and Applications of Glass. Elsevier, Amsterdam, p. 266, 1980).

The refractive index of a medium is the ratio of the speed of light in a vacuum to its speed in the medium, and is the square root of the relative permittivity of the medium at that frequency. When measured with visible light, the refractive index is related to the electronic polarizability of the medium. Solvents with high refractive indexes, such as aromatic solvents, should be capable of strong dispersion interactions. Unlike the other measures described here, the refractive index is a property of the pure liquid without the perturbation generated by the addition of a probe species. 

In Equation 12.13, N is the number density of molecules in the beam of radiation (and is thus inversely proportional to the molar volume, Vm), and o is the permittivity of the vacuum. A useful and widely employed method to evaluate the sum in Equation 12.13 leads via the closure approximation to a one-term equation commonly known as the dispersion relation,... 

Theoretical treatment of the femtosecond pulse propagation in a bulk Kerr medium with the dispersion of dielectric permittivity was based on the... 

Ya.B. applied formal perturbation theory to the interaction of an atom with the electrons of a metal, where the latter are assumed to be free. Meanwhile, Casimir and Polder and Lifshitz neglected the spatial dispersion of the dielectric permittivity of the metal. Therefore, in the region of small distances, frequencies of order ui0 are important at small distances in the sense indicated above, as are arbitrarily small frequencies at large distances. In both limits the dielectric permittivity of the metal is not at all close to one. Meanwhile, the perturbation theory used by Ya.B. corresponds formally to an expansion in powers of e - 1. and is therefore not applicable in this case. Neglecting the spatial dispersion is valid, however, only at distances r > a (a is the Debye radius in the metal) of the atom from the surface. At the opposite extreme, r a, the wave vectors kj 1/r > a vF/u>0 Me of importance (vF is the electron speed at the Fermi boundary). In this region of strong spatial dispersion perturbation theory can be applied, and the (--dependence satisfies Zeldovich s law. 

Dispersions of nanoparticles in ferroelectric liquid crystals (FLCs) predominantly focused on induced or altered electro-optic effects, but also on the alignment of FLCs. Raina and co-workers reported on a gradual decrease of the dielectric permittivity, e, by doping with SiC>2 nanoparticles at frequencies up to 1 kHz and a rather minor increase of as well as an increase in optical transmission at frequencies above 2 kHz [279]. Liang et al. used BaTiC>3 nanoparticles (31 nm in diameter after grinding commercially available 90 nm nanoparticles Aldrich) and showed, perhaps expectably, a twofold increase in the spontaneous polarization... 

If the concentration of an electrolyte is rather low, then at least three elementary types of motion stipulate dispersion of the complex permittivity in this polar fluid ... 

Figure 50. Calculated contributions of the ionic permittivity for the imaginary (a, c) and real (b, d) parts of the total complex permittivity (solid lines) dashed lines refer to the calculation, neglecting the ionic dispersion, (a, b) For NaCl-water solution (c, d) for KCl-water solution. Cm = 0.5 mol/liter, Tion/r = 10.

The dispersion of the dielectric response of each contribution leads to dielectric losses of the matter which can be mathematically expressed by a complex dielectric permittivity ... 

However, the temperature, at which the maximum of the initial scattered light occurs, seems to be related to the scattering angle 9S and thus to the period Ag , respectively. Figure 9.14(b) shows the correspondence between the temperature Tm of maximum intensity Ig and the spatial period Agn. A spatial disorder of the smallest polar structures occurs at Tm = 45 °C, while the spatial orientation of the largest structures remains stable up to Tm = 60 °C. Such big dispersion of the thermal decay of polar structures over Agn unambiguously illustrates the relaxor behavior of sbn. At the same time it is a key point to understand the bandwidth in the determination of the phase transition temperature Tm in sbn from different methods. For example, in sbn doped with 0.66 mol% Cerium, Tm detected from the maximum of the dielectric permittivity e at 100 Hz (e-method) equals Tm = 67 °C [20], Determination of Tm from the inflection point of the spontaneous electric polarization P3... 

For the closed description of the electron transfer in polar medium, it is necessary to express the reorganization energy in the formula (27) via the characteristics of polar media (it is assumed that the high-temperature approach can be always applied to the outer-sphere degrees of freedom). It was done in the works [12, 19, 23], and most consistently in the work of Ovchinnikov and Ovchinnikova [24] where the frequency dependence of the medium dielectric permittivity e(co) is taken into account exactly, but the spatial dispersion was neglected. 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

Eq. (4), frequency-dependent, such that the limit for a(w) in Eq. (8) becomes physically acceptable. Under conditions appropriate to the correct limit, the normalized real and imaginary parts of the complex permittivity and the normalized dielectric conductivity take on the form depicted in Fig. (1). Here, is the relaxation time in the limit of zero frequency (diabatic limit). Irrespective of the details of the model employed, both a(w) and cs(u>) must tend toward zero as 11 + , in contrast to Eq. (8), for any relaxation process. In the case of a resonant process, not expected below the extreme far-infrared region, a(u>) is given by an expression consistent with a resonant dispersion for k (w) in Eq. (6), not the relaxation dispersion for K (m) implicit in Eq. 

The dielectric behaviour of pure water has been the subject of study in numerous laboratories over the past fifty years. As a result there is a good understanding of how the complex permittivity t = E — varies with frequency from DC up to a few tens of GHz and it is generally agreed that the dielectric dispersion in this range can be represented either by the Debye equation or by some function involving a small distribution of relaxation times. 

To determine the nature of the dispersion curve at frequencies between the main dispersion and resonance regions the data contained in references (4-15) at 20 C were combined with the 70GHz values and the permittivity results of Asfar and Hasted up to frequencies less than 2THz. The combined data were then fitted to the equation... 

The high frequency limit of for this second process is therefore n. The result of the fit is shown in Table III where the mean values of the various parameters and their associated 95% confidence intervals are given. Considering the small amplitude of the second dispersion both in absolute t rms and in relation to the main dispersion the parameters 6m, n and Y are quite well defined, and therefore it may be concluded that the double Debye representation is an acceptable description of the dielectric behaviour of water up to around 2THz. Other alternative interpretations are clearly possible but no attempt has been made here to follow these up at this stage. What is clear is that a small subsidiary dispersion region in the far infrared is necessary to account for all the presently available permittivity data, and that such a dispersion is centred around 650GHz and has an amplitude of about 2.4 in comparison with that of the principal dispersion which is approximately 75. 

Solutions of many proteins, synthetic polypeptides, and nucleic acids show large increases in permittivity c (u>) over that of solvent, normally aqueous, at sufficiently low frequencies f = w/2ir of steady state AC measurements, but with dispersion and absorption processes which may lie anywhere from subaudio to megahertz frequencies. Although our interest here is primarily in counterion fluctuation effects as the origin of polarization of aqueous DNA solutions, we first summarize some relevant results of other models for biopolymers. 

In Tung s experiments, aqueous solutions of calf thymus DNA from commercial sources with average molecular weights in the range 106 to 107 were measured from 10 Hz to several kHz with or without added salts (NaCl and BaCl.) Some of the results are shown in Figure 1. In Figure l(a)f dispersions of e (real part of e ) at 5, 15, and 25C extrapolated to zero frequency show that the static permittivities increase with increasing temperature, which is contrary to the l/T dependence expected for permanent dipole polarization, as in eq 7, but can be explained as a consequence of polyion polarization. 

Observed dispersions e1 for four sized fractions with average molecular weights M from 0.44 to 1.85 x 106 Dalton at concentrations C 8 to 9 x 10 5 are shown in Figure 1(b). Within experimental Error, the values of Ae /C, where Ac is the increment of static permittivity for added solute, were independent of concentration but increased linearly with molecular weight, as predicted by the McTague-Gibbs theory for molecular weights below about 106 Dalton. 

The present evidence is thus that kinetic effects may account for half or more of permittivity decreases of ionic solutions and this may be an important factor in determing the amplitude of the Y dispersion in conducting biopolymer solutions and lead to revisions in estimated nature and amount of bound water. The effect may also have some bearing on dielectric properties of cell interiors and membranes if these have appreciable conductances. It would seem premature to attempt definitive answers to such questions until the relative importance of static and kinetic effects in presumably simpler ionic solutions has been better established experimentally in comparison with theory which treats them self-consistently. 

B. Protein Solutions. The dielectric properties of proteins and nucleic acids have been extensively reviewed (10, 11). Protein solutions exhibit three major dispersion ranges. One occurs at RF s and is believed to arise from molecular rotation in the applied electric field. Typical characteristic frequencies range from about 1 to 10 MHz, depending on the protein size. Dipole moments are of the order of 200-500 Debyes and low-frequency increments of dielectric permittivity vary between 1 and 10 units/g protein/100 ml of solution. The high-frequency dielectric permittivity of this dispersion is lower than that of water because of the low dielectric permittivity of the protein leading to a high-frequency decrement of the order of 1 unit/g protein/... 

The commonest case to have been analysed is that of 3-0 systems. James Clerk Maxwell (1831-1879) deduced that the permittivity m of a random dispersion of spheres of permittivity ei in a matrix of relative permittivity e2 is given by... 

Figure 2.38 illustrates that in the case of an ionic solid the optical mode of the lattice vibration resonates at an angular frequency, co0, in the region of 1013Hz. In the frequency range from approximately 109-10nHz dielectric dispersion theory shows the contribution to permittivity from the ionic displacement to be nearly constant and the losses to rise with frequency according to... 

Figure 4.53 shows the dispersion spectra of hydrated Ca-HC, K-HC, and Na-HC at 300K [120], They are plotted in the x-axis, and the natural logarithm of the frequency (Hz) versus the natural logarithm of the real part of the relative permittivity in the y-axis. This experiment shows once more the higher mobility of monovalent cations in comparison with divalent cations and the higher mobility of Na+ with respect to K+. The cause of this effect is due to the inferior cationic radius of Na+ in comparison with that of K+. 

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